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Kurt Gödel

Kurt Gödel
Kurt Friedrich Gödel (/ˈkɜrt ɡɜrdəl/; German: [ˈkʊʁt ˈɡøːdəl] ( ); April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher. Considered with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell,[1] A. N. Whitehead,[1] and David Hilbert were pioneering the use of logic and set theory to understand the foundations of mathematics. Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. Life[edit] Childhood[edit] In his family, young Kurt was known as Herr Warum ("Mr. Studying in Vienna[edit] Related:  mathematiciensMathematics

David Hilbert David Hilbert (German: [ˈdaːvɪt ˈhɪlbɐt]; January 23, 1862 – February 14, 1943) was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces,[3] one of the foundations of functional analysis. Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Life[edit] Hilbert remained at the University of Königsberg as a Privatdozent (senior lecturer) from 1886 to 1895. The Mathematical Institute in Göttingen. The Göttingen school[edit] Among Hibert's students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. In English:

One Hundred Interesting Mathematical Calculations, Number 9: Archive Entry From Brad DeLong's Webjournal One Hundred Interesting Mathematical Calculations, Number 9 One Hundred Interesting Mathematical Calculations, Number 9: False Positives Suppose that we have a test for a disease that is 98% accurate: if one has the disease, the test comes back "yes" 98% of the time (and "no" 2% of the time), and if one does not have the disease, the test comes back "no" 98% of the time (and "yes" 2% of the time). Suppose further that 0.5% of people--one out of every two hundred--actually has the disease. Your test comes back "yes." Suppose just for ease of calculation that we have a population of 10000, of whom 50--one in every two hundred--have the disease. If you test "no" you can be very happy indeed: there is only one chance in 9752 that you are the unlucky guy who had the disease and yet tested negative. If you test "yes" you are less happy. From John Allen Paulos's Innumeracy .

Gödel numbering A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects. Simplified overview[edit] Gödel noted that statements within a system can be represented by natural numbers. In simple terms, we devise a method by which every formula or statement that can be formulated in our system gets a unique number, in such a way that we can mechanically convert back and forth between formulas and Gödel numbers. Gödel's encoding[edit] Gödel used a system based on prime factorization. Example[edit] Lack of uniqueness[edit]

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔr/ KAN-tor; German: [ˈɡeɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfɪlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918[1]) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive – even shocking – that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré[3] and later from Hermann Weyl and L. The harsh criticism has been matched by later accolades. Life[edit] Youth and studies[edit] Cantor, ca. 1870. Teacher and researcher[edit] In 1867, Cantor completed his dissertation, on number theory, at the University of Berlin. ...

FIBONACCI "...considering both the originality and power of his methods, and the importance of his results, we are abundantly justified in ranking Leonardo of Pisa as the greatest genius in the field of number theory who appeared between the time of Diophantus [4th century A.D.] and that of Fermat" [17th century] R.B. McClenon [13]. [Numbers in square brackets refer to REFERENCES at the end of this article.] 1. During the twelfth and thirteenth centuries, many far-reaching changes in the social, political and intellectual lives of people and nations were taking place. By the end of the twelfth century, the struggle between the Papacy and the Holy Roman Empire had left many Italian cities independent republics. Among these important and remarkable republics was the small but powerful walled city-state of Pisa which played a major role in the commercial revolution which was transforming Europe. Pisa today is best known for its leaning tower (inclined at an angle of about 161/2o to the vertical).

Spirasolaris Carl Friedrich Gauss Johann Carl Friedrich Gauss (/ɡaʊs/; German: Gauß, pronounced [ɡaʊs]; Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician who contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, mechanics, electrostatics, astronomy, matrix theory, and optics. Sometimes referred to as the Princeps mathematicorum[1] (Latin, "the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest mathematician since antiquity," Gauss had an exceptional influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.[2] Early years[edit] Gauss was a child prodigy. The year 1796 was most productive for both Gauss and number theory. Middle years[edit] Gauss, who was 24 at the time, heard about the problem and tackled it. One such method was the fast Fourier transform. Later years and death[edit] Religious views[edit]

Fibonacci number A tiling with squares whose side lengths are successive Fibonacci numbers An approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34. Animated GIF file showing successive tilings of the plane, and a graph of approximations to the Golden Ratio calculated by dividing successive pairs of Fibonacci numbers, one by the other. Uses the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 In mathematics, the Fibonacci numbers or Fibonacci sequence are the numbers in the following integer sequence: or (often, in modern usage): (sequence A000045 in OEIS). By definition, the first two numbers in the Fibonacci sequence are 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two. In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation or

List of topics named after Leonhard Euler - Wikipedia, the free ... In mathematics and physics, there are a large number of topics named in honor of Leonhard Euler, many of which include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Unfortunately, many of these entities have been given simple and ambiguous names such as Euler's function, Euler's equation, and Euler's formula. Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. Euler's conjectures[edit] Euler's equations[edit] Euler's formulas[edit] Euler's functions[edit] Euler's identities[edit] Euler's identity e iπ + 1 = 0.Euler's four-square identity, which shows that the product of two sums of four squares can itself be expressed as the sum of four squares.Euler's identity may also refer to the pentagonal number theorem. Euler's numbers[edit] Euler's theorems[edit] Euler's laws[edit] Other things named after Euler[edit] Topics by field of study[edit] Graph theory[edit] Music[edit]

Sacred geometry As worldview and cosmology[edit] The belief that God created the universe according to a geometric plan has ancient origins. Plutarch attributed the belief to Plato, writing that "Plato said God geometrizes continually" (Convivialium disputationum, liber 8,2). In modern times the mathematician Carl Friedrich Gauss adapted this quote, saying "God arithmetizes".[2] At least as late as Johannes Kepler (1571–1630), a belief in the geometric underpinnings of the cosmos persisted among scientists. Closeup of inner section of the Kepler's Platonic solid model of planetary spacing in the Solar system from Mysterium Cosmographicum (1596) which ultimately proved to be inaccurate Natural forms[edit] Art and architecture[edit] Geometric ratios, and geometric figures were often employed in the design of Egyptian, ancient Indian, Greek and Roman architecture. In Hinduism[edit] Unanchored geometry[edit] Music[edit] See also[edit] Notes[edit] Further reading[edit] External links[edit] Sacred geometry at DMOZ

Carl Friedrich Gauss Un article de Wikipédia, l'encyclopédie libre. Carl Friedrich Gauss Portrait de Johann Carl Friedrich Gauss (1777-1855), réalisé par Christian Albrecht Jensen Signature Johann Carl Friedrich Gauß (prononcé en allemand [gaʊs] La qualité extraordinaire de ses travaux scientifiques était déjà reconnue par ses contemporains. Gauss dirigea l'Observatoire de Göttingen et ne travailla pas comme professeur de mathématiques – d'ailleurs il n'aimait guère enseigner – mais il encouragea plusieurs de ses étudiants, qui devinrent d'importants mathématiciens, notamment Gotthold Eisenstein et Bernhard Riemann. Biographie[modifier | modifier le code] En 1792, le duc de Brunswick remarque ses aptitudes et lui accorde une bourse afin de lui permettre de poursuivre son instruction. Tombe de Gauss au Albanifriedhof de Göttingen. Il est élu le 12 avril 1804 membre de la Royal Society. La fille de Gauss, Therese (1816—1864). En 1810, il se remarie avec « Minna » Waldeck (4 août 1810).